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Theorem difdif 4093
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (𝐴 ∖ (𝐵𝐴)) = 𝐴

Proof of Theorem difdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.45im 826 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
2 iman 405 . . . . 5 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
3 eldif 3929 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
42, 3xchbinxr 338 . . . 4 ((𝑥𝐵𝑥𝐴) ↔ ¬ 𝑥 ∈ (𝐵𝐴))
54anbi2i 625 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
61, 5bitr2i 279 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
76difeqri 4087 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  cdif 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-dif 3922
This theorem is referenced by:  dif0  4315  undifabs  4409
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